via orthogonal polynomials

MATHEMATICS OF COMPUTATIONVOLUME 53, NUMBER 187JULY 1989, PAGES 249-264

A Fast Algorithm for Rational Interpolation

Via Orthogonal Polynomials

By Ömer Egecioglu*and Çetin K. Koç**

Abstract. A new algorithm for rational interpolation is proposed. Given the data

set, the algorithm generates a set of orthogonal polynomials by the classical three-

term recurrence relation and then uses Newton interpolation to find the numerator

and the denominator polynomials of the rational interpolating function. The number

of arithmetic operations of the algorithm to find a particular rational interpolant is

0(N2), where N + 1 is the number of data points. A variant of this algorithm that

avoids Newton interpolation can be used to construct all rational interpolants using

only 0(N2) arithmetic operations.

1. Introduction. Let R(m, n) be the set of rational functions of the form

rm,n(x) = Pm(x)/qn(x), where pm(x) and qn(x) are polynomials of degree m and

n, respectively. If a set of N +1 = m + n + 1 pairs of points (x¿, fi) for 0 < i < N is

given, then the rational interpolation problem is defined as the task of determining

a rational function rmj„(x) € R(m,n) such that

(1.1) rm,n(xi) = fi forO<i<N,

where the x¿'s are distinct elements. We also assume that the /j's are the values of

a function f(x) at the nodes x¿ for all i.

In the case of polynomial interpolation (i.e., n = 0) it is always possible to

construct a unique polynomial satisfying (1.1), but this is not true for rational

interpolation. For fixed m and n, a rational interpolant that satisfies (1.1) at all

points may not exist, even though it may be possible to satisfy (1.1) on a subset

of the given point set by means of an rll¡u(x) e R(p,u) with p < m and v < n.

For this particular pair m and n, the point set becomes a degenerate configuration.

The points (xj,fj) for which

rli,v\Xj) ^p Jj

are called the unattainable points. Roughly speaking, the existence of an unattain-

able point (xj,fj) for 0 < j < N means that Xj is a common zero of the numerator

and the denominator polynomials. Thus a degenerate point set contains one or

more unattainable points.

A direct algorithm for rational interpolation solves a set of linear equations

(1-2) Pm(^t)-/l<7n(xl) = 0, 0<i<N,

Received June 3, 1987; revised February 4, 1988.

1980 Mathematics Subject Classification (1985 Revision). Primary 65D05; Secondary 33A65.

Key words and phrases. Rational interpolation, orthogonal polynomials, Hankel matrices.

'Supported in part by NSF Grant No. DCR-8603722."Supported in part by Lawrence Livermore National Laboratory Contract No. LLNL-7526225

while the author was with the Department of Electrical and Computer Engineering, University of

California, Santa Barbara, California 93106.

©1989 American Mathematical Society0025-5718/89 $1.00 + $.25 per page

249

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

250 ÖMER EGECIOGLU AND ÇETIN K. KOÇ

to compute the coefficients of the polynomials pm(x) and qn(x), the oldest method

being an elegant elimination process due to Jacobi [6, pp. 326-330], [9]. Jacobi's

method yields exact determinantal formulae for the coefficients of the denominator

and numerator polynomials. A more recent algorithm by Schneider and Werner

[10] makes use of the barycentric representation of the rational interpolant. Here

a set of linear equations of size n is solved by applying Gaussian elimination to

compute the coefficients of qn(x).

Direct methods can be computationally inefficient because the solution of a gen-

eral set of linear equations may require as much as 0(t73) arithmetic operations.

Jacobi's algorithm, on the other hand, results in a Hankel system of linear equa-

tions, which in turn can be solved with 0(n2) arithmetic operations.

We remark that alongside the direct algorithms there exists an array of iterative

methods to construct rm,„(x), and the reader is referred to [4] for a review of these

algorithms.

In this paper we propose a fast direct algorithm for rational interpolation. Given

the data set (x¿,/¿) for 0 < i < N, the algorithm first generates a sequence qo(x),

qi(x),... ,qn(x) of orthogonal polynomials on the discrete set {xo,xi,... , x;v} with

respect to certain weights. The values of these polynomials at the nodes x¿ for

0 < i < N are computed by using the classical three-term recurrence relation sat-

isfied by orthogonal polynomials. Once the values of a particular polynomial are

known, its coefficients can be computed easily via Newton interpolation. The last

orthogonal polynomial qn(x) generated by this algorithm is the required denom-

inator polynomial of the rational interpolant. Once the denominator polynomial

qn(x) is determined, the construction of the numerator pTO(x) becomes a polyno-

mial interpolation problem. We prove that the number of arithmetic operations of

this algorithm to find a particular rational interpolant for given values of m and n

is 0(N2).

The coefficients of the orthogonal polynomials can also be found without the

application of the Newton interpolation algorithm. This is again achieved by using

the three-term recurrence formula. The resulting algorithm requires 0(N2) arith-

metic operations to generate the polynomials qo(x), qi(x),... ,q^(x). If the same

reasoning is applied to the data (x¿, f~~l) for 0 < i < N, the coefficients as well as

the values of the numerator polynomials po(x), Pi(x),... ,pn(x) can be computed.

We show that in this way all rational interpolants rm,n(x) with m + n = N and

0 < 77 < N can be computed using a total of only 0(N2) arithmetic operations

provided that fi ^0 for all 0 < i < N, and the orthogonal polynomials exist.

The outline of this paper is as follows: In Section 2, we describe the Jacobi

algorithm for rational interpolation. Section 3 outlines the use of orthogonal poly-

nomials to compute a particular rational interpolant based on a generalization of

Jacobi's approach and Newton interpolation. Fast computation of all rational inter-

polants that avoids Newton interpolation is described in Section 4. This is followed

by examples for each one of these algorithms in Section 5, and conclusions and

suggestions for further research in Section 6.

2. Jacobi Rational Interpolation Algorithm. Let [g(x)]0...N denote the

Nth divided difference of a function g(x) with respect to the node values x¿ for


A FAST ALGORITHM FOR RATIONAL INTERPOLATION 251

0 < i < N. Put

(2.1) w(x) = (x - x0)(x - xi) • • ■ (x - xjv)

andTV

(2.2) Wi = w'(xi) = Y[{xi-Xj).

3=0JTti

By direct application of the Lagrange interpolation formula one has

(2-3) [ff(*)W = ££

where gi = g(xi). Note that (2.3) gives the leading coefficient of the polynomial

of degree at most N that takes the values & at the node points x¿ for 0 < i <

N. Clearly, if g(x) is a polynomial of degree strictly less than A^, then [g(x)]o-.-N

vanishes.

From (1.2) we can write for any j > 0

(2.4) APm(xi) = x3lfiqn(xi) [mQ^{^N

Wi Wi

and by summing these terms over i we obtain

y> x{pm(xi) _ y* xifiqn(xi)

¿-i Wi -nwii=0 1=0

Using the definition (2.3) this can be simplified as

(2.5) [x3pm(x)}0...N = [x3f(x)qn(x)}0-N-

The left-hand side of (2.5) is equal to the Nth divided difference of the polynomial

x3prn(x). Thus, if j is in the range 0 < j < n — 1, we have

deg(xJpm(x)) =j + m<n-l + m = N-l.

Hence,

(2.6) [xjpm(x)]o...N =0 for 0 < j < n - 1

and therefore

(2.7) [x3f(x)qn(x)}o...N = 0 forO<j<n-l.

The equation (2.7) allows us to compute the coefficients of qn(x) by solving a set

of linear equations. Since there are n equations and n + 1 unknowns, one of the

parameters can be chosen arbitrarily.

If qn(x) is represented in the standard power basis,

n

Qn(x) = J2bl<xk>k=0

then from (2.7) the 6/t's satisfy the following linear system of equations

£^^ = £^¿ fc = 0i o^y<«-if^—' Wi *—' Wi ^-^i=0 ' i=0 * fc=0


252 OMER EGECIOGLU AND ÇETIN K. KOÇ

which can be put in the form

n

(2.8) J2 hJ+kbk = 0,k=0

where

(2-9)

N

hs = Ei=0

fixlWi

0<j<n-l,

0 < s < 277 - 1.

The system of equations given in (2.8) is a Hankel system. As indicated before,

we have one degree of freedom in choosing the coefficients of qn (x) in any rational

interpolation. From now on we will assume that qn(x) is a monk polynomial of

degree n, that is bn = 1- With this choice, (2.8) can be rewritten as the matrix

equation

(2.10)

Therefore,

(2.11)

where

hohi

h2

hih2

h3

hh3

tl4

hn-i

h„

hn+i

hn-i h„ hn

Qn(x)det(Hn-i)

+ 1 ■•' l2n-2

h0 hi

hi h2

hn-i hn

1 x

bo

hb2

bn-]

hn

hn+l

hn+2

72n-l

det

hn-i hn

hn hn+i

h-2n-2 h2n-l„n-1 „.n

Hj =

ho hihi h2

hjhj+i

hj+ i h2]

As it stands, the determinantal formula (2.11) is computationally inefficient to

construct the denominator polynomial qn(x). However, Trench [12] has provided a

fast algorithm to solve the Hankel system (2.10) directly. If all of the matrices Hq,

Hi, ..., Hn-i are nonsingular, then the number of arithmetic operations to invert

Hn-i is proportional to n2. Thus the system in (2.10) can be solved in 0(n2)

arithmetic operations given the nonsingularity condition on the principle minors of

Hn-i-

Note that the existence of the denominator of degree n for rm,n(x) depends

only on the nonsingularity of the matrix Hn-i by (2.11). The nonsingularity of

all the matrices Ho, Hi,... ,Hn-i is thus equivalent to the existence of all of the

denominator polynomials qo(x), qi(x),... ,qn(x). It turns out that this is precisely

the assumption required on the matrices H0, Hi,..., Hn-i for the construction of

orthogonal polynomials outlined in the next section.

Once the coefficients of qn(x) are found, we can evaluate qn(x) at all of the node

points to check if the data set is degenerate for the given values of m and n and



thus locate unattainable points. The coefficients of pm (x) can then be computed by

performing a polynomial interpolation for the data set (x¿, fiqn(xi)) for 0 < i < m.

To summarize, Jacobi's algorithm for rational interpolation can be carried out

in the following manner:

Jacobi Algorithm

Input: (xi, fi) for 0 < i < N.

Output: Coefficients of pm(x) and qn(x) of the rational interpolant rm,n(x).

Step 1. Compute Wi for 0 < i < N using (2.2).

Step 2. Compute hs for 0 < s < 2n — 1 using (2.9).

Step 3. Solve the resulting Hankel system by using the Trench algorithm to find

the coefficients of qn (x) in the standard form.

Step 4. Check whether or not the data set is degenerate for the given values of m

and n, i.e., check if qn(xt) = 0 for any 0 < i < N by evaluating qn(x) at all x¿ and

thus locating the unattainable points.

Step 5. Interpolate the data set (x¿, fiqn(xi)) for 0 < i < m using the Newton

interpolation algorithm to find pm (x) in the Newton form.

Thus we have

THEOREM 1. Given the data set (xj,/¿) for 0 < i < N, the Jacobi algorithm

computes the coefficients of the rational interpolant rm,n(x) which satisfies (1.1)

using 0(N2) arithmetic operations.

Proof. We will count the number of arithmetic operations at each step of the

algorithm.

In Step 1, Wi can be calculated with N subtractions and N — 1 multiplications for

any fixed i. For all i = 0,1..., N this clearly takes 0(N2) arithmetic operations.

In Step 2, first Xs is computed for all s = 0,1,... ,2n —1 with 2n-2 multiplications.

To compute hs we perform N +1 multiplications, N + Í divisions and TV additions.

For all s = 0,1,..., 2n - 1 this takes O(Nn) operations. Thus Step 2 takes 0(N2)

arithmetic operations.

The solution of the Hankel system takes 0(n2) arithmetic operations as stated

earlier. Step 4 consists of evaluating a polynomial of degree n at N + 1 points.

Hence up to Step 5, the number of operations required is no more than 0(N2).

Finally, in Step 5 we apply the Newton interpolation to construct the polynomial

Pm(x) which takes |m(m-f 1) = 0(7772) = 0(N2) operations [5]. Thus the number

of operations of the Jacobi algorithm add up to 0(N2). D

It should be noted that Step 3 of the Jacobi algorithm can be carried out with

0(n2) arithmetic operations without the assumption that the principal minors of

Hn-i be nonzero (except det(Hn-i)) by applying an algorithm of Rissanen to solve

Hankel and Toeplitz systems [8]. Nevertheless, if one is interested in constructing

all rm,n (x) for m + n = A7 and 0 < n < N, we see that the construction of all of

the denominator polynomials qo(x), qi(x),..., qn(x) would require the solution of n

such systems, resulting in an 0(n3) algorithm. Given the nonsingularity assumption

on the matrices Hq, Hi,..., Hn-i, we can compute all of the polynomials qo(x),



qi(x),... ,qn(x) with a total of 0(n2) arithmetic operations by using orthogonal

polynomials. This is the subject matter of the next section.

Alternatively, the process that yielded the coefficients of qn(x) can be repeated

with the data set (x¿,/t_1) for 0 < i < N to construct the coefficients of pm(x). To

this end, note that (1.2) can be written as

Qn(Xi) - TPm(xi) =0, 0<i<N,Ji

provided /¿ # 0 for all 0 < i < N. By representing pm (x) in its standard form

TO

Pm(x) = ^2akxk

k=0

and applying the same reasoning as in the Eqs. (2.4) through (2.8), we obtain

TO

(2.12) ^2 h3+kak = °» 0 < j < m - 1,

where

(2.13) fe's==£-£L 0<S<2m-l.-_n JiWi

k=0

N

t=0

Again the system of equations in (2.12) becomes a Hankel system. By assuming

Pm(x) to be a monic polynomial for the moment, we obtain a system of linear

equations of size m with the unknowns ao, a\,... ,am-i similar to (2.10):

(2.14)

KK

l/4-1

K h'2h'2 h'3

W, h'A

"m "*n

h'm-l

h'm

nm+l

/i2to-2-I Lam-1

a0

0-2

nm+l

hm+2

K,m+1 "2to-2-I L-m-ij L«2m-1-

Since Pm(x) and qn(x) cannot be forced to be monic at the same time, we

need to multiply one of these polynomials with a nonzero constant to make them

consistent with (1.1). Thus if qn(x) and pm(x) are monic polynomials as solutions

of the equations (2.10) and (2.14), the corresponding rational interpolant is simply

(x)(2.15)

where

i,n(x) —

am =

OmPn

Qn(x)

foQn(xo)

Pm(xo)

assuming that (xq, /o) is an attainable point. If we put

(2.16) *3 =

h'o

h\

K

K

h'J+i

h>3 '

h'j+i

h'0"3 "¿+1 ''2j J

and use the Trench algorithm to solve (2.14), we see that the matrices H'Q, H[,...,

H'm_1 should be nonsingular.



Thus, Newton interpolation can be avoided in Step 5 of the Jacobi Algorithm by

solving another Hankel system to determine pm(x). Since the number of arithmetic

operations to solve a Hankel system or to perform a Newton interpolation is the

same, 0(m2), we see that this approach does not provide any gains in terms of

number of arithmetic operations. However, we will make use of this idea in Section

4 to devise an algorithm to construct all rational interpolants using only 0(N2)


The Jacobi algorithm may have undesirable numerical properties because of the

stability issues involved in the way the divided differences (i.e., the individual hs's)

are calculated. The algorithm proposed by Schneider and Werner in [10] makes

use of the barycentric representation of the rational interpolant and seems to have

better numerical properties. On the other hand, there is a trade-off in terms of

time complexity, since the linear system of equations that arise in this particular

algorithm has no additional structure to facilitate a reduction in the overall time

complexity. In particular, the use of Gaussian elimination to solve a general system

of linear equations, as proposed, requires 0(n3) operations. Thus the number of

arithmetic operations required by the Schneider-Werner algorithm to compute a

particular rational interpolant is 0(N3) because of this apparent bottleneck, as

opposed to 0(N2) achieved by the Jacobi algorithm.

3. Rational Interpolation Using Orthogonal Polynomials. Our point

of departure will be the Jacobi algorithm and we will show that it is possible

to compute the denominator polynomial qn (x) without solving a system of linear

equations.

We start by noting that any polynomial of degree j in x¿ can be used as a mul-

tiplier in Eq. (2.4). It follows that (2.5) and (2.7) hold for an arbitrary polynomial

tj (x) degree j in place of x3. Hence we have

N fi(3.1) [tj(x)f(x)qn(x)]0-N = ^2 —tji^Q^Xi) = 0

¿=o Wl

for 0 < j < n — 1. Set c¿ = fi/wi for 0 < i < N and define a discrete symmetric

bilinear form {•, •) on the space of polynomials of degree less than or equal to n by

setting

N

(3.2) (tj(x),tk(x)) := Y,Citj(xi)tk(xi).i=0

Using (3.2), the linear system (3.1) can be written as

JV

(3.3) ^2cit3(xi)qn(xi) = (tj(x),qn(x)) = 0, 0<j<n-l.i=0

Note that the bilinear form (3.2) does not necessarily define an inner product

since it is possible to have (tj(x),tj(x)) < 0. For our purposes, however, it suffices

to assume nondegeneracy, that is, (tj(x), tj(x)) ^ 0 for 0 < j < n. If we require the

set {io(x), ii(x),..., tn(x)} to be orthogonal with respect to this discrete bilinear

form, then the nondegeneracy condition together with (3.3) imply that qj(x) is

a constant multiple of tj(x) for j = 0,1,... ,n. In particular, the denominator

polynomials qo(x), qi(x),... ,qn(x) are orthogonal with respect to (3.2).



The nondegeneracy of the bilinear form (3.2) is in turn guaranteed by the non-

singularity of the matrices Ho, Hi,..., Hn-i-

THEOREM 2. There exists a sequence of orthogonal polynomials {<o(x),

ti(x),... ,£n(x)} on the set {xo,xi,... ,x^} with respect to the symmetric bilin-

ear formN

fi(tj(x),tk(x)) = ^2-^-tj(Xi)tk(xl)

i=0

if and only if the matrices Hq, Hi, ..., Hn-i are all nonsingular.

Proof. From the general theory of orthogonal polynomials [1], the polynomial

tj (x) is given explicitly (up to a constant multiple) by the determinant

(3.4)

Note that

tj(x) = det

(1,1) <l,x)

(x, 1) (x,x)

iy-M) <x^-\x)1 X

(i,x»;>

(x,x})

(x3~l,x3)

xi

Nh(xk,xl) = J2^k+l = hk+l.

i=o —

Thus the coefficient of x1 in tj(x) is precisely the determinant of H,_i, which has

to be nonzero. D

Since we have assumed that qn(x) is monic, by making use of (3.4) we arrive at

Jacobi's explicit formula for the denominator polynomials given in (2.11).

The conventional method for generating orthogonal polynomials is the Gram-

Schmidt orthogonalization process. However in our case, this approach would re-

quire 0(n2) inner product operations to generate qn(x). A more efficient technique

is to use the three-term recurrence relation for orthogonal polynomials to generate

the values of the polynomials q-, (x) directly. With this approach, the total number of

inner product operations required to compute the values qo(xi), qi (x¿),..., gn_i (x¿)

for any given point x¿ becomes only 0(n) [3], [7], [11].

More precisely, let {to(x), ti(x),..., tn(x)} be a set of polynomials satisfying the

orthogonality relationship with respect to the weights c¿ and the sequence of data

points Xi for 0 < i < N. It is well known that

(3.5)i_i(x) = 0, i0(x) = l,

tj+i(x) = (X - Oj)tj(x) - ßjtj-i(x), j = 0,1,...,

where a} and ßj are constants determined as

_ (xtj(x),t3(x)) a _ (t:i(x),t:i(x))(3.6) a,

(tj(x),tj(x)) 'A =

(tj-l(x),tj-i(x))'

Note that the relation (3.5) generates monic orthogonal polynomials, and requires

only the nondegeneracy of the underlying symmetric bilinear form.



Let T = [Tji] be the (N + 1) x (N + 1) matrix where

(3.7) Tji = tj(xi), 0<i,j<N.

In other words, the jth row of T consists of the vector of values of the jth orthogonal

polynomial tj(x) at the nodes Xo, Xi,...,xjv- The following procedure generates

the first n + 1 rows of the matrix T using the three-term recursion (TTR) in (3.5).

Procedure TTR.

Input: n and (x,, /¿) for 0 < i < N.

Output: T = [Tji] for 0 < j < n and 0 < i < N.

Step 1. Compute Wi and c¿ for 0 < i < N using

tt fiWi = JJ(x¿ - Xj) and c¿ = —.

3=0

Step 2. Set T0i = 1 for 0 < i < N, ß0 = 0, and compute

N N

O; anH n/r. =

0oio = ^2 c»x*' 9° = ^2Ci' and a°= "^

i=0 i=0

Step 3. Set Ti¿ = x¿ - ao for 0 < i < N, and compute

N N

li = ̂ 2 CiXiT2t, ei = J2ciTu, and aj = J-, ft = £¿=o ¿

5iep 4. For 1 < j < n - 1 compute

-i 0o '

ri+M = (xt - a^Tyi - ^Tj-a.i, 0 < i < N,N N

l3 + l = J2 CiX*Tj+l,i> e3 + l = ^2 C'T3+hi't=0 t=0

- li±l R - e3 + 1a3 + l - ~d-' P3 + Í - -J—-

i/j + 1 0j

We will denote by T: = TTR(n, a, fi) the (n +1) x (N +1) matrix produced by

TTR from the input data 77 and (x¿, /¿), 0 < i < N.

LEMMA 1. Given the data set (x¿,/¿) for 0 < i < N, and n < N, Procedure

TTR computes the first n + 1 rows of the matrix T using

(3.8) 2AT2 + lONn + 2N + 10n-l = 0(N2)

arithmetic operations. In particular, all rows of T can be computed with

(3.9) 12N2 + 12N-1 = 0(N2)




Proof. In Step 1 the computation of w¿ takes 2N — 1 operations for a particular

value of z. All 0 < i < N takes (N+1)(2N—1) operations. The c¿'s can be computed

in N + 1 operations. Hence Step 1 of the procedure takes 2N2 + 2N arithmetic

operations. The computation of ao in Step 2 takes 2>N + 2 arithmetic operations.

In Step 3, Tu and Tx2 are computed with 2(N +1) arithmetic operations. Then to

compute ai and ßi we need to perform 5N + 5 arithmetic operations. Thus Step

3 takes IN + 7 operations. In Step 4 for a particular value of j the computation

of Tj+iti and lf+li takes b~N + 5 arithmetic operations. Then to compute aJ+i

and ßj+i, 57V + 5 operations are needed. For all 1 < j < n — 1, Step 4 takes

(n - 1)(107V + 10) operations. The result in (3.8) thus follows. To find (3.9) we

replace n with N in (3.8). D

We are now in a position to describe the rational interpolation algorithm via

orthogonal polynomials and to prove that rational interpolation can be done using

0(N2) arithmetic operations. The following algorithm first generates the first n+1

rows of the matrix Q = [Qji], where Qji = Oj(x¿), and then applies the Newton

interpolation algorithm to find qn(x) and pm(x).

Algorithm 1.

Input: (xj, fi) for 0 < i < N.

Output: Coefficients of pm(x) and qn(x) of the rational interpolant rm<n(x).

Stepl. Q: = TTR(n,xl,fi).

Step 2. Interpolate the data set (xi,Qni) for 0 < i < n to compute the coefficients

of qn(x) in the Newton form by using the Newton interpolation algorithm.

Step 3. Compute fiQn% for all 0 < i < N and then interpolate the data set

(xí, fiQni) for 0 < i < 777 to compute the coefficients of pm(x) in the Newton

form by using the Newton interpolation algorithm.

THEOREM 3. Given the data set (xi,ft) for 0 < i < N, Algorithm 1 computes

the coefficients of the rational interpolant rmyn(x) which satisfies (1.1) using 0(N2)


Proof. By Lemma 1, Step 1 takes 2N2 + lOAn + 2N + lOn - 1 arithmetic opera-

tions. Step 2 is an interpolation process with n + 1 points. This requires |n(n + 1)

operations [5]. Similarly, in Step 3 we first compute fiQni for all 0 < i < m and

then perform a polynomial interpolation with m+1 points; hence, 7n+l+|m(7n+l)

arithmetic operations need to be performed in this step. If we sum the number of

operations at each step we conclude that Algorithm 1 takes

(3.10) 2/V2 + §(n2 + m2) + lOAn + 2/V + f n + \m = 0(N2)

arithmetic operations. D

One remarkable property of Algorithm 1 is that it makes it trivial to check

whether the given data set is degenerate for any values of m and n with m + n = N.

To check for degeneracy, we compute the values of Qj% for all 0 < j < N in

Step 1. This implies that the input set for the Procedure TTR is N and (x¿,/,)

for 0 < i < N. The number of arithmetic operations to compute all rows Q is

12N2 + 12N - 1 as given in (3.9). Thus the total number of arithmetic operations



of Algorithm 1 increases slightly to

(3.11) 12/V2 + |(n2 + m2) + 12A + fn+|7n,

which is still 0(N2). This allows us to select a particular value of m and n for

which the data set is not degenerate while keeping the total number of operations

within 0(N2).

4. Fast Computation of All Rational Interpolants. Note that it is possible

to compute the coefficients of the polynomial qn(x) recursively using the three-term

recurrence formula in (3.5) and thus avoid the Newton interpolation in Step 2 of

Algorithm 1. This can be done by applying the three-term recursion directly to the

coefficients of the denominator polynomials. More precisely, let B = [Bjk] be the

(N + 1) x (N + 1) matrix in which the jth row consists of the coefficients of the

polynomial qj(x), i.e.,

(4.1) q,(x) = ¿ Bjkxk.

fc=o

Then S is a lower triangular matrix with unit diagonal whose elements satisfy the

recursion

(4.2) Bj+ltk = Bjik-! - ajBjk - ßjBj-i,k, 0<k<j<N-l,

induced by (3.5). In (4.2) we take

Bj-i=0 for 0 < j'< TV,4.3 h -J- y

H_i,fc = 0 forO<k<N.

Thus using this recursion formula we can generate the coefficients of the polyno-

mials qj(x) for 0 < j < n. The values of the polynomials qj(x) at the node points

Xi (i.e., Qji) are computed at each step to calculate aj and ßj, but this also helps

to locate the unattainable points if the point set happens to be degenerate for the

particular values of m and n.

If only one rational interpolant is needed then, in Step 2 of Algorithm 1, the

choice between the Newton interpolation algorithm or the application of recurrence

formula in (4.2) is somewhat arbitrary, since both algorithms will require 0(n2)

arithmetic operations. For the Newton algorithm, however, the constant in front

of the order is smaller.

More importantly, the generation of the coefficients of the polynomials qj(x),

0 < j < n, in 0(n2) arithmetic operations suggests a drastic cut-down on the

number of arithmetic operations when all rational interpolants rm,n(x) for 0 < n <

N with m = N - n are computed. We note that for 0 < j < N the coefficients

of the polynomials pj (x) can also be computed similarly by applying the recursion

in (4.2) to the data (x¿, /,_1) for 0 < i < N, as we already remarked at the end of

Section 2. This is possible provided /¿ / 0 for 0 < z < TV and H0, H[,...,H'N_l

are nonsingular. This given, define A = [Ajk] and P = [Pij] to be (N + 1) x (N + 1)

matrices where the jth row of A contains the coefficients of the polynomial Pj(x),

(4.4) p,(x) = ¿Ajfcxfc,

k=0


260 ÖMER EGECIOGLU AND ÇETIN K. KOÇ

and the jth row of P contains the values of the polynomial Pj(x) at the nodes x¿

Pji = pj(xi), 0<i,j<N,

similar to the matrices B and Q.

The following algorithm first generates the values and the coefficients of the

polynomials qj(x) for all 0 < j < N. The algorithm then proceeds to generate the

values and the coefficients of the polynomials pj (x) for 0 < j < N by applying the

same technique to the data (x¿, ff ) for 0 < i < N.

Algorithm 2.

Input: (xí, fi) for 0 < i < N.

Output: Coefficients of pm(x) and qn(x) for 0 < n < TV and m = N — n.

Stepl. Q:=TTR(N,Xi,fi).Step 2. Set Bjj = 1 for 0 < j < N and Bjk = 0 for 0 < j < k < N. Set Bw = -a0.

For all 0 < k < j < N - 1 compute

Bj+i,k — Pj,k-\ — OLjBjk — ßjBj-i^k-

Step 3. Compute f~l for 0 < i < N and P := TTR(N, Xi,f~l).

Step 4. Set Ajj = 1 for 0 < j < N and Ajk = 0for0<j<k<N. Set Aw = -a0.

For all 0 < k < j < N - 1 compute

Aj+i,k = Aj,k-i - oijAjk - ßjAj-itk-

Step 5. Update the coefficients of Pj(x) according to (2.15):

Ajk = !(ß^-Ajk, 0<k<j<N.fjo

At the end of Algorithm 2, the coefficients of the polynomials

are Amk and Bnk, respectively, for 0 < k < N. Note that Amk =

Bnk = 0 for k > n.

THEOREM 4. Algorithm 2 computes all rational interpolants rmin(x) for

n + m = N and n = 0,1,... ,N using 0(N2) arithmetic operations provided the

matrices Ho, Hi,..., Hjv-i o.nd H'Q, H[,..., H'N_1 are nonsingular.

Proof. As we showed in Theorem 2, the orthogonal polynomials qo(x), qi(x),...,

o;v(x) can be constructed if and only if the matrices Ho, Hi,...,Hjv-i are all

nonsingular. By applying the same technique to the data (x¿, /,_1) for 0 < i < N,

we conclude that the polynomials po(x), pi(x),... ,Pn(x) can be constructed if the

matrices H0, H[,..., H'N_1 are all nonsingular and if fi ^ 0 for all 0 < i < N.

To compute the number of arithmetic operations required for Algorithm 2 we

count the operations at each step. Step 1 takes 12Ar2+12Ar-l arithmetic operations

by Lemma 1. Also Step 3 will take N +1 + 12N2 + 12N - 1 arithmetic operations.

For each of Step 2 and 4 notice that the recursion in (4.2) requires X)i_Ti 12í=o ^ =

2N2 + 27V - 4 arithmetic operations. Also in Step 5 only the lower triangular part

of the matrix A is updated which results in

N-l j j .

2N + 2+^2^2l = -N2 + -N + 3j = l fc=0

pm(x) and gn(x)

0 for k > m and



arithmetic operations. Thus Algorithm 2 requires a total of

(4.5) Ç/V2 + 6iAr_6 = 0(Ar2)2 2

arithmetic operations to compute all rational interpolants. D

5. Examples. The algorithms have been implemented on a VAX-11/780 com-

puter running Unix 4.2 BSD, using the Pascal programming language. Even though

we have not yet conducted a detailed experimental study of the numerical prop-

erties of these algorithms, we present computer generated solutions to two simple

interpolation problems.

Example 1 (Algorithm 1). Given f(x) = \x\, find r2,2(x) which interpolates f(x)

at the node points —1, -0.5, 0, 0.5, 1. We apply Step 1 of Algorithm 1 to the data

and obtain the following Q matrix of size (5x5):

Q =

1.00

1.00

1.50

0.75

0.00

1.00 1.00

0.50 0.00

0.75 0.50

0.75 0.00

0.00 0.25

1.00

0.50

0.75

0.75

0.00

1.00

1.00

0.75-0.75

0.00

The rows of the matrix Q are the values of the denominator polynomials qj(x) at

the nodes x¿ for 0 < i, j < 4. As mentioned before, if Qn% = 0 for an i then

the point (xj,/¿) is an unattainable point for the (m,n) pair. An inspection of all

entries in Q gives the following result:

m n unattainable points

none

(0,0)none

(0,0)0 4 all points except (0,0)

The rational interpolant 7-2,2(x) interpolates f(x) at all points. In order to deter-

mine q2(x), we use the Newton interpolation algorithm. The values of the polyno-

mial q2 (x) at the node points are seen from the third row to be

q2(-l) = 1.50, q2(-0.5) = 0.75, q2(0) = 0.50, g2(0.5) = 0.75, q2(l) = 0.75.

Since three points are necessary and sufficient to determine the coefficients of 0.2 (x),

we choose the first three points: (-1, 1.50) (—0.5, 0.75), (0,0, 0.5), and find

q2(x) = x2+0.5. In order to determine the coefficients of the numerator polynomial,

first we compute

/0Q20 = 150, AQ21 = 0.375, /2Q22 = 0.00

and then apply Newton interpolation to the data (-1, 1.50), (-0.5, 0.375), (0, 0.00)

to find the numerator polynomial P2(x) = 1.50x2. Thus,

p2(x) 1.50X2r2,2(x) = —7—r = ,,-.-•

q2(x) x2 +0.5



Here we note that since f2 = 0, Algorithm 2 cannot be applied to find all rational

interpolants for this problem.

Example 2 (Algorithm 2). Given f(x) = 2X, find all rational interpolants of

f(x) at the node points —2, —1, 0, 1, 2. The application of Step 1 and Step 2 of

Algorithm 2 to this data set produces the following Q and B matrices:

Q =

B

48-192

384

1-6

26-72

96

1-7

36-120

192

0

1-9

38-66

1-6

26-72

96

0

0

1-9

23

1-5

18-42

48

0

0

0

0

1

1-4

12-24

24

Thus we observe that for this interpolation problem all points are attainable for all

(m, n) pairs. The elements of B are the coefficients of the denominator polynomials

<7j(x) for 0 < j < 4. In other words, the entries in the first column are the coefficient

of x° in qj(x), the entries in the next column are the coefficient of x, and so on.

The unit diagonal entries correspond to the coefficient of the leading term x3 in

qj(x).

Similarly, the application of Steps 3, 4 and 5 of Algorithm 2 produces the coef-

ficients of the numerator polynomials, namely the A matrix:

A =

96-72

26-6

1

0-12

9_ IS

61116

0

0

1_ 3

42396

0

0

0_1_12J_16

0

00

0J_96 J

Hence the rational interpolants are found to be

Po(x)»"0,4 (X) =

rlAx) =

r2,2 (x) =

r3,l(z) =

r<i,o(x)

96

94 (x)

Pi(x)

«73 (z)

P2(X)

Q2(x)

PA1)Qi(x)

Pi(x)

Qo(x)

6x3 + 23x2

12x + 72

66x + 96'

-x3 + 9x2-38x + 72'

x2 + 9x + 26

x2-9x + 26'

x3 + 9x2 + 38x + 72

-12X + 72

x4 + 6x3 + 23x2 + 66x + 96

96

6. Summary and Conclusions. We have described two algorithms for ra-

tional interpolation. Given that the Hankel matrices Ho, Hi,...,Hv-i are all

nonsingular, the first one of these algorithms (Algorithm 1) generates orthogonal

polynomials qo(x), <7i(x),... ,<7jv(x) which are the monic denominator polynomials



of the associated rational interpolants. If the computation of the corresponding nu-

merator polynomials are then carried out by Newton interpolation, then the number

of arithmetic operations required becomes 0(7V3). However, if the corresponding

matrices H'Q, H[,..., H'N_1 for the numerators are also nonsingular, then Newton

interpolation can be avoided, and all rational interpolants can be found with only

0(N2) arithmetic operations. This is the content of Algorithm 2.

Note that in case a particular matrix Hj_i happens to be singular, then it is no

longer possible to continue the generation of the denominator polynomials qj(x),

(7i+i(x),... ,9jv(x) by using the three-term recursion, as the necessary condition

for the existence of the desired orthogonal family of polynomials is not satisfied.

Thus, if the discrete symmetric bilinear form we are interested in happens to be

degenerate, it is not clear how to proceed with the method of orthogonal poly-

nomials to generate the rest of the rational interpolants. At this point, however,

the remaining denominator polynomials can be computed individually by repeated

application of Rissanen's algorithm at a cost of 0(N2) operations each. Similarly,

the singularity of a matrix H',_l associated with a numerator polynomial puts a

limit on the applicability and the operating range of Algorithm 2. Nevertheless, it

is possible to combine these two approaches for a hybrid algorithm whose running

time is guaranteed to be no worse than the existing algorithms for rational inter-

polation. In addition, both Algorithm 1 and Algorithm 2 provide simple provisions

to check for degeneracy of the interpolation problem at hand.

Although the proposed algorithms are fast, an extensive analysis of their prac-

ticality in terms of numerical stability has not yet been carried out. It should be

noted however, that both Algorithm 1 and Algorithm 2 can be parallelized on ei-

ther shared-memory or message-passing multiprocessor models. This aspect of the

algorithms will be reported elsewhere [2].

Acknowledgment. We are grateful to the anonymous referee whose comments

and suggestions have greatly improved the readability of this paper.

Department of Computer Science

University of California

Santa Barbara, California 93106

E-mail: [email protected]

Department of Electrical Engineering

University of Houston

Houston, Texas 77204

E-mail: [email protected]

1. P. J. DAVIS, Interpolation and Approximation, Dover, New York, 1975.

2. Ö. EGECIOGLU k Ç. K. KOC, An 0(N log N) Parallel Algorithm for Rational Interpolation,Technical Report No. TRCS88-2, Department of Computer Science, University of California, Santa

Barbara, August 1988.

3. G. E. FORSYTHE, "Generation and use of orthogonal polynomials for data fitting with a

digital computer," J. Soc. Indust. Appl. Math., v. 5, 1957, pp. 74-78.

4. P. R. GRAVES-MORRIS k T. R. HOPKINS, "Reliable rational interpolation," Numer. Math.,

v. 36, 1981, pp. 111-128.5. F. B. HILDEBRAND, An Introduction to Numerical Analysis, McGraw-Hill, New York, 1956.

6. T. MUIR, The Theory of Determinants, vol. 2, Dover, New York, 1960.



7. A. RALSTON k P. RABINOWITZ, A First Course in Numerical Analysis, McGraw-Hill, New

York, 1978.

8. J. RlSSANEN, "Solution of linear equations with Hankel and Toeplitz matrices," Numer.

Math., v. 22, 1974, pp. 361-366.9. T. J. RlVLIN, An Introduction to the Approximation of Functions, Dover, New York, 1969.

10. C. SCHNEIDER k W. WERNER, "Some new aspects of rational interpolation," Math. Comp.,

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via orthogonal polynomials

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